I'm using left-over building materials for an outbuilding.

Can I use galvanized drip edge with copper roofing? Or will that cause some sort of galvanic corrosion?

zinc + copper + acid rain = galvanic corrison

Quoted: I'm using left-over building materials for an outbuilding. Can I use galvanized drip edge with copper roofing? Or will that cause some sort of galvanic corrosion?

Very likely.  While not as rapid as in Saltwater or being submerged in freshwater like a boat; you do live in a fairly humid and rainy environment ( as compared to my state ) so it would still deteriorate portions of the outbuilding fairly quickly.

What if, say, the galvanized drip edge was put down first, then a layer of ice-and-rain shield, then the copper roofing? Would the runoff from the copper onto the drip edge still cause corrosion even if the two metals weren't actually touching?

Is there any way you can insulate your drip edge from the roofing by using say, tarpaper?

If there is any moisture in the adjoining surfaces sufficient current can be passed to cause corrosion.

Paint the hell out of the drip edge with a primer and epoxy paint from a spray can.

Do an internet search on "galvanic series" for more information.  In general terms, the closer two elements or alloys are in the galvanic series, the better for corrosion resistance - you don't want mating parts at the extreme ends of the series without a good strategy to insulate the parts from each other.

www.corrosionsource.com/handbook/galv_series.htm